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# 請幫我解這題高微題目

Let R be a regular region in R3 with piecewise smooth boundary.

Show that the volume of R is 1/3 ∫∫ R F*n dA where F( x , y , z ) = Xi + Yj + Zk.

我已經知道答案為3A，但非常需要過程，謝謝

Update:

修正一下，是 ∫ ∫ R(下標) F*n dA

Update 2:

抱歉，我搞錯了，這題是證明題，答案不是3A

### 1 Answer

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- 天助Lv 71 decade agoFavorite Answer
F=(x,y,z), divF=1+1+1=3

Let S be the boundary of the region R(volume of R is V),then

∫∫_S F‧n dA=∫∫∫_R divF dv

=∫∫∫_R 3dv=3∫∫∫_R dv=3V

so, V=(1/3)∫∫_S F‧n dA

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